Unoriented bordism
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$$ where $\tau$ is the involution mapping $(x,[y]) $ to $(-x, [\bar y])$ and $\bar y = (\bar y_0,...,\bar y_n)$ for $y =(y_0,...y_n)$. These manifolds are now cold Dold manifolds. | $$ where $\tau$ is the involution mapping $(x,[y]) $ to $(-x, [\bar y])$ and $\bar y = (\bar y_0,...,\bar y_n)$ for $y =(y_0,...y_n)$. These manifolds are now cold Dold manifolds. | ||
− | Using the results by Thom | + | Using the results by Thom {{cite|Thom1954}} Dold shows that these manifolds give ring generators of $\mathcal N_*$. |
− | {{beginthm|Theorem (Dold) | + | {{beginthm|Theorem (Dold) {{cite|Dold1956}}:}} For $i$ even set $x_i:= [P(i,0) ]= [\mathbb {RP}^i]$ and for $i = 2^r(2s+1)-1$ set $x_i:=[ P(2^r-1,s2^r)]$. Then for $i \ne 2^k-1$ |
$$ | $$ | ||
x_2,x_4,x_5,x_6,x_8,... | x_2,x_4,x_5,x_6,x_8,... |
Revision as of 16:31, 8 January 2010
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Contents |
1 Introduction
We denote the non-oriented bordism groups by . The sum of these groups
are a ring under cartesian products of the manifolds. Thom [Thom] has shown that this ring is a polynomial ring over in variables for and he has shown that for even one can take for . Dold \cite {Dold} has constructed manifolds for with odd.
2 Construction and examples
Dold constructs certain bundles over with fibre denoted by
Using the results by Thom [Thom1954] Dold shows that these manifolds give ring generators of .
Theorem (Dold) [Dold1956]: 2.1. For even set and for set . Then for
are polynomial generators of olver :
3 Invariants
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4 Classification/Characterization (if available)
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5 Further discussion
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6 References
- [Dold1956] A. Dold, Erzeugende der Thomschen Algebra , Math. Z. 65 (1956), 25–35. MR0079269 (18,60c) Zbl 0071.17601
- [Thom] Template:Thom
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502
This page has not been refereed. The information given here might be incomplete or provisional. |